System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Percentage Accurate: 61.1% → 94.5%

Time: 14.4s

Alternatives: 9

Speedup: 28.3×

• could not determine a ground truth

  1. x = 4.591225699928997e+227
  2. y = 1.462898921944362e+23
  3. z = 2.4151047119286998e+24
  4. t = -1.435487445349641e+43

Specification

Math

\[\begin{array}{l} \\ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log(((1.0 - y) + (y * Math.exp(z)))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log(((1.0 - y) + (y * math.exp(z)))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (log(((1.0 - y) + (y * exp(z)))) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log(((1.0 - y) + (y * Math.exp(z)))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log(((1.0 - y) + (y * math.exp(z)))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (log(((1.0 - y) + (y * exp(z)))) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.118:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+145}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -0.118)
   (- x (/ (log (fma (expm1 z) y 1.0)) t))
   (if (<= y 4.9e+145)
     (- x (* (/ (expm1 z) t) y))
     (-
      x
      (/
       (log (fma (fma (* (fma 0.16666666666666666 z 0.5) y) z y) z 1.0))
       t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -0.118) {
		tmp = x - (log(fma(expm1(z), y, 1.0)) / t);
	} else if (y <= 4.9e+145) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(fma((fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -0.118)
		tmp = Float64(x - Float64(log(fma(expm1(z), y, 1.0)) / t));
	elseif (y <= 4.9e+145)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(fma(Float64(fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -0.118], N[(x - N[(N[Log[N[(N[(Exp[z] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e+145], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.11799999999999999

   1. Initial program 50.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot \left(e^{z} - 1\right)\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(y \cdot \left(e^{z} - 1\right) + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{\left(e^{z} - 1\right) \cdot y} + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(e^{z} - 1, y, 1\right)\right)}}{t} \]

      4. lower-expm1.f64 93.6

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(z\right)}, y, 1\right)\right)}{t} \]

   5. Applied rewrites 93.6%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}{t} \]

   if -0.11799999999999999 < y < 4.90000000000000003e145

   1. Initial program 68.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

         \[\leadsto x - y \cdot \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 98.6

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 98.6%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 4.90000000000000003e145 < y

   1. Initial program 2.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right)\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) \cdot z} + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right), z, 1\right)\right)}}{t} \]

      4. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) + y}, z, 1\right)\right)}{t} \]

      5. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) \cdot z} + y, z, 1\right)\right)}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y, z, y\right)}, z, 1\right)\right)}{t} \]

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6}, y \cdot z, \frac{1}{2} \cdot y\right)}, z, y\right), z, 1\right)\right)}{t} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{z \cdot y}, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{z \cdot y}, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      10. lower-*.f64 90.1

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, \color{blue}{0.5 \cdot y}\right), z, y\right), z, 1\right)\right)}{t} \]

   5. Applied rewrites 90.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, 0.5 \cdot y\right), z, y\right), z, 1\right)\right)}}{t} \]

   6. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right)\right)}}{t} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) \cdot z} + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right), z, 1\right)\right)}}{t} \]

      4. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) + y}, z, 1\right)\right)}{t} \]

      5. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) \cdot z} + y, z, 1\right)\right)}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y, z, y\right)}, z, 1\right)\right)}{t} \]

      7. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\left(z \cdot y\right)} + \frac{1}{2} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      8. associate-*r* N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot z\right) \cdot y} + \frac{1}{2} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      9. distribute-rgt-in N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot z + \frac{1}{2}\right)}, z, y\right), z, 1\right)\right)}{t} \]

      10. +-commutative N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot z\right)}, z, y\right), z, 1\right)\right)}{t} \]

      11. *-commutative N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot z\right) \cdot y}, z, y\right), z, 1\right)\right)}{t} \]

      12. lower-*.f64 N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot z\right) \cdot y}, z, y\right), z, 1\right)\right)}{t} \]

      13. +-commutative N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot z + \frac{1}{2}\right)} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      14. lower-fma.f64 90.1

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, z, 0.5\right)} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

   8. Applied rewrites 90.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(\left(1 - y\right) + y \cdot e^{z}\right) \leq 100:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (log (+ (- 1.0 y) (* y (exp z)))) 100.0)
   (- x (* (/ (expm1 z) t) y))
   (* (- x) -1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (log(((1.0 - y) + (y * exp(z)))) <= 100.0) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = -x * -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.log(((1.0 - y) + (y * Math.exp(z)))) <= 100.0) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = -x * -1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.log(((1.0 - y) + (y * math.exp(z)))) <= 100.0:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = -x * -1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) <= 100.0)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(Float64(-x) * -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 100.0], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[((-x) * -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < 100

   1. Initial program 55.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

         \[\leadsto x - y \cdot \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 91.3

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 91.3%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 100 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))))

   1. Initial program 88.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(-x\right) \cdot \color{blue}{\left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

   5. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\frac{\mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t \cdot x} - 1\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(-x\right) \cdot -1 \]
   7. Step-by-step derivation

   8. Applied rewrites 59.1%

      \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+156}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+145}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.4e+156)
   (- x (/ (log (fma z y 1.0)) t))
   (if (<= y 4.9e+145)
     (- x (* (/ (expm1 z) t) y))
     (-
      x
      (/
       (log (fma (fma (* (fma 0.16666666666666666 z 0.5) y) z y) z 1.0))
       t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.4e+156) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else if (y <= 4.9e+145) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(fma((fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.4e+156)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	elseif (y <= 4.9e+145)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(fma(Float64(fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.4e+156], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e+145], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.39999999999999994e156

   1. Initial program 50.4%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(y \cdot z + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{z \cdot y} + 1\right)}{t} \]

      3. lower-fma.f64 61.1

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   5. Applied rewrites 61.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   if -1.39999999999999994e156 < y < 4.90000000000000003e145

   1. Initial program 65.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

         \[\leadsto x - y \cdot \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 93.4

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 93.4%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 4.90000000000000003e145 < y

   1. Initial program 2.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right)\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) \cdot z} + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right), z, 1\right)\right)}}{t} \]

      4. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) + y}, z, 1\right)\right)}{t} \]

      5. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) \cdot z} + y, z, 1\right)\right)}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y, z, y\right)}, z, 1\right)\right)}{t} \]

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6}, y \cdot z, \frac{1}{2} \cdot y\right)}, z, y\right), z, 1\right)\right)}{t} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{z \cdot y}, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{z \cdot y}, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      10. lower-*.f64 90.1

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, \color{blue}{0.5 \cdot y}\right), z, y\right), z, 1\right)\right)}{t} \]

   5. Applied rewrites 90.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, 0.5 \cdot y\right), z, y\right), z, 1\right)\right)}}{t} \]

   6. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right)\right)}}{t} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) \cdot z} + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right), z, 1\right)\right)}}{t} \]

      4. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) + y}, z, 1\right)\right)}{t} \]

      5. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) \cdot z} + y, z, 1\right)\right)}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y, z, y\right)}, z, 1\right)\right)}{t} \]

      7. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\left(z \cdot y\right)} + \frac{1}{2} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      8. associate-*r* N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot z\right) \cdot y} + \frac{1}{2} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      9. distribute-rgt-in N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot z + \frac{1}{2}\right)}, z, y\right), z, 1\right)\right)}{t} \]

      10. +-commutative N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot z\right)}, z, y\right), z, 1\right)\right)}{t} \]

      11. *-commutative N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot z\right) \cdot y}, z, y\right), z, 1\right)\right)}{t} \]

      12. lower-*.f64 N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot z\right) \cdot y}, z, y\right), z, 1\right)\right)}{t} \]

      13. +-commutative N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot z + \frac{1}{2}\right)} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      14. lower-fma.f64 90.1

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, z, 0.5\right)} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

   8. Applied rewrites 90.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+156}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+145}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.4e+156)
   (- x (/ (log (fma z y 1.0)) t))
   (if (<= y 4.9e+145)
     (- x (* (/ (expm1 z) t) y))
     (- x (/ (log (fma (fma (* (* 0.16666666666666666 z) y) z y) z 1.0)) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.4e+156) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else if (y <= 4.9e+145) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(fma(((0.16666666666666666 * z) * y), z, y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.4e+156)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	elseif (y <= 4.9e+145)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(fma(Float64(Float64(0.16666666666666666 * z) * y), z, y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.4e+156], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e+145], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(N[(0.16666666666666666 * z), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.39999999999999994e156

   1. Initial program 50.4%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(y \cdot z + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{z \cdot y} + 1\right)}{t} \]

      3. lower-fma.f64 61.1

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   5. Applied rewrites 61.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   if -1.39999999999999994e156 < y < 4.90000000000000003e145

   1. Initial program 65.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

         \[\leadsto x - y \cdot \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 93.4

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 93.4%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 4.90000000000000003e145 < y

   1. Initial program 2.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right)\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) \cdot z} + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right), z, 1\right)\right)}}{t} \]

      4. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) + y}, z, 1\right)\right)}{t} \]

      5. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) \cdot z} + y, z, 1\right)\right)}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y, z, y\right)}, z, 1\right)\right)}{t} \]

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6}, y \cdot z, \frac{1}{2} \cdot y\right)}, z, y\right), z, 1\right)\right)}{t} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{z \cdot y}, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{z \cdot y}, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      10. lower-*.f64 90.1

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, \color{blue}{0.5 \cdot y}\right), z, y\right), z, 1\right)\right)}{t} \]

   5. Applied rewrites 90.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, 0.5 \cdot y\right), z, y\right), z, 1\right)\right)}}{t} \]

   6. Taylor expanded in z around inf

      \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot \left(y \cdot z\right), z, y\right), z, 1\right)\right)}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.1%

      \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y, z, y\right), z, 1\right)\right)}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+156}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+145}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, 0.5, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.4e+156)
   (- x (/ (log (fma z y 1.0)) t))
   (if (<= y 4.9e+145)
     (- x (* (/ (expm1 z) t) y))
     (- x (/ (log (fma (fma (* z y) 0.5 y) z 1.0)) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.4e+156) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else if (y <= 4.9e+145) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(fma((z * y), 0.5, y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.4e+156)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	elseif (y <= 4.9e+145)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(fma(Float64(z * y), 0.5, y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.4e+156], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e+145], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(z * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.39999999999999994e156

   1. Initial program 50.4%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(y \cdot z + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{z \cdot y} + 1\right)}{t} \]

      3. lower-fma.f64 61.1

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   5. Applied rewrites 61.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   if -1.39999999999999994e156 < y < 4.90000000000000003e145

   1. Initial program 65.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

         \[\leadsto x - y \cdot \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 93.4

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 93.4%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 4.90000000000000003e145 < y

   1. Initial program 2.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot \left(y + \frac{1}{2} \cdot \left(y \cdot z\right)\right)\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot \left(y + \frac{1}{2} \cdot \left(y \cdot z\right)\right) + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{\left(y + \frac{1}{2} \cdot \left(y \cdot z\right)\right) \cdot z} + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(y + \frac{1}{2} \cdot \left(y \cdot z\right), z, 1\right)\right)}}{t} \]

      4. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \left(y \cdot z\right) + y}, z, 1\right)\right)}{t} \]

      5. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\left(y \cdot z\right) \cdot \frac{1}{2}} + y, z, 1\right)\right)}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y \cdot z, \frac{1}{2}, y\right)}, z, 1\right)\right)}{t} \]

      7. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot y}, \frac{1}{2}, y\right), z, 1\right)\right)}{t} \]

      8. lower-*.f64 90.1

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot y}, 0.5, y\right), z, 1\right)\right)}{t} \]

   5. Applied rewrites 90.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, 0.5, y\right), z, 1\right)\right)}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 82.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\mathsf{fma}\left(0.5, z, 1\right)}{t} \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 0.0) (* (- x) -1.0) (- x (* (* (/ (fma 0.5 z 1.0) t) z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = -x * -1.0;
	} else {
		tmp = x - (((fma(0.5, z, 1.0) / t) * z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-x) * -1.0);
	else
		tmp = Float64(x - Float64(Float64(Float64(fma(0.5, z, 1.0) / t) * z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[((-x) * -1.0), $MachinePrecision], N[(x - N[(N[(N[(N[(0.5 * z + 1.0), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 80.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(-x\right) \cdot \color{blue}{\left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\frac{\mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t \cdot x} - 1\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(-x\right) \cdot -1 \]
   7. Step-by-step derivation

   8. Applied rewrites 65.2%

      \[\leadsto \left(-x\right) \cdot -1 \]

   if 0.0 < (exp.f64 z)

   1. Initial program 49.4%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

         \[\leadsto x - y \cdot \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 87.5

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 87.5%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   6. Taylor expanded in z around 0

      \[\leadsto x - \left(z \cdot \left(\frac{1}{2} \cdot \frac{z}{t} + \frac{1}{t}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 88.7%

      \[\leadsto x - \left(\frac{\mathsf{fma}\left(0.5, z, 1\right)}{t} \cdot z\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+156} \lor \neg \left(y \leq 4.9 \cdot 10^{+145}\right):\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.4e+156) (not (<= y 4.9e+145)))
   (- x (/ (log (fma z y 1.0)) t))
   (- x (* (/ (expm1 z) t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.4e+156) || !(y <= 4.9e+145)) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.4e+156) || !(y <= 4.9e+145))
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.4e+156], N[Not[LessEqual[y, 4.9e+145]], $MachinePrecision]], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.39999999999999994e156 or 4.90000000000000003e145 < y

   1. Initial program 31.4%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + y \cdot z\right)}}{t} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(y \cdot z + 1\right)}}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\color{blue}{z \cdot y} + 1\right)}{t} \]

      3. lower-fma.f64 72.7

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   5. Applied rewrites 72.7%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y, 1\right)\right)}}{t} \]

   if -1.39999999999999994e156 < y < 4.90000000000000003e145

   1. Initial program 65.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

         \[\leadsto x - y \cdot \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 93.4

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 93.4%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+156} \lor \neg \left(y \leq 4.9 \cdot 10^{+145}\right):\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+31}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.4e+31) (* (- x) -1.0) (fma (- y) (/ z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.4e+31) {
		tmp = -x * -1.0;
	} else {
		tmp = fma(-y, (z / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.4e+31)
		tmp = Float64(Float64(-x) * -1.0);
	else
		tmp = fma(Float64(-y), Float64(z / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.4e+31], N[((-x) * -1.0), $MachinePrecision], N[((-y) * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.39999999999999982e31

   1. Initial program 79.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(-x\right) \cdot \color{blue}{\left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\frac{\mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t \cdot x} - 1\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(-x\right) \cdot -1 \]
   7. Step-by-step derivation

   8. Applied rewrites 65.1%

      \[\leadsto \left(-x\right) \cdot -1 \]

   if -2.39999999999999982e31 < z

   1. Initial program 50.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y \cdot \left(e^{z} - 1\right)}{t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

         \[\leadsto x - y \cdot \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)} \]

      3. *-commutative N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y} \]

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 87.2

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 87.2%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot z}{t}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{t}\right)\right)} + x \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \frac{z}{t}}\right)\right) + x \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{z}{t}} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(y\right), \frac{z}{t}, x\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, \frac{z}{t}, x\right) \]

      7. lower-/.f64 88.1

         \[\leadsto \mathsf{fma}\left(-y, \color{blue}{\frac{z}{t}}, x\right) \]

   8. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, \frac{z}{t}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.1% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) \cdot -1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (- x) -1.0))

C

double code(double x, double y, double z, double t) {
	return -x * -1.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -x * (-1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return -x * -1.0;
}

Python

def code(x, y, z, t):
	return -x * -1.0

Julia

function code(x, y, z, t)
	return Float64(Float64(-x) * -1.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -x * -1.0;
end

Wolfram

code[x_, y_, z_, t_] := N[((-x) * -1.0), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
2. Add Preprocessing

3. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)\right)} \]

   2. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

   4. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right) \]

   5. lower--.f64 N/A

      \[\leadsto \left(-x\right) \cdot \color{blue}{\left(\frac{\log \left(\left(1 + y \cdot e^{z}\right) - y\right)}{t \cdot x} - 1\right)} \]

5. Applied rewrites 78.0%

   \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\frac{\mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t \cdot x} - 1\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \left(-x\right) \cdot -1 \]
7. Step-by-step derivation

8. Applied rewrites 70.9%

   \[\leadsto \left(-x\right) \cdot -1 \]
9. Add Preprocessing

Developer Target 1: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5}{y \cdot t}\\ \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{t\_1}{z \cdot z}\right) - t\_1 \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- 0.5) (* y t))))
   (if (< z -2.8874623088207947e+119)
     (- (- x (/ t_1 (* z z))) (* t_1 (/ (/ 2.0 z) (* z z))))
     (- x (/ (log (+ 1.0 (* z y))) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -0.5d0 / (y * t)
    if (z < (-2.8874623088207947d+119)) then
        tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0d0 / z) / (z * z)))
    else
        tmp = x - (log((1.0d0 + (z * y))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (Math.log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -0.5 / (y * t)
	tmp = 0
	if z < -2.8874623088207947e+119:
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)))
	else:
		tmp = x - (math.log((1.0 + (z * y))) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5) / Float64(y * t))
	tmp = 0.0
	if (z < -2.8874623088207947e+119)
		tmp = Float64(Float64(x - Float64(t_1 / Float64(z * z))) - Float64(t_1 * Float64(Float64(2.0 / z) / Float64(z * z))));
	else
		tmp = Float64(x - Float64(log(Float64(1.0 + Float64(z * y))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -0.5 / (y * t);
	tmp = 0.0;
	if (z < -2.8874623088207947e+119)
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	else
		tmp = x - (log((1.0 + (z * y))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-0.5) / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.8874623088207947e+119], N[(N[(x - N[(t$95$1 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[(2.0 / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(1.0 + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -288746230882079470000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (- x (/ (/ (- 1/2) (* y t)) (* z z))) (* (/ (- 1/2) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t))))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))